Is 'wat' Pali for 'temple', and 'vihara' Pali for monastery? I hear these terms and then those are the translations, but I don't know which language they are. If they're not Pali, are they common place throughout the Theravada world?

"wat" is Thai I think, though "vihara" is a term in many Indic dialects, including what we now call "Pali". It come from the verb "to dwell" or "to abide", viharati. Originally, a vihara was a dwelling, including the physical sense, eg. a hut or the like, but also the more abstract types of meditative dwelling, eg. sunnata vihara, "dwelling in emptiness". Later, it becomes more and more used as the physical dwelling, particularly for any sort of renunciants (not just meditators), and hence "monastery" in English.

My recently moved Blog, containing some of my writings on the Buddha Dhamma, as well as a number of translations from classical Buddhist texts and modern authors, liturgy, etc.: Huifeng's Prajnacara Blog.